# Analytical Models of Velocity, Reynolds Stress and Turbulence Intensity in Ice-Covered Channels

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## Abstract

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## 1. Introduction

## 2. Material and Methods

#### 2.1. Vertical Distribution of Longitudinal Velocity

#### 2.2. Vertical Distribution of Reynolds Stress

#### 2.3. Vertical Distribution of Turbulence Intensity

## 3. Experimental Verification

## 4. Model Parameters

#### 4.1. ${m}_{b}$ and ${m}_{i}$

#### 4.2. ${n}_{b}$ and ${n}_{i}$

#### 4.3. ${K}_{0}$

## 5. Results and Discussion

#### 5.1. Model Verification

#### 5.2. Discussion

#### 5.2.1. Manning’s Coefficients ${n}_{b}$ and ${n}_{i}$

#### 5.2.2. Flow Parameters ${m}_{b}$ and ${m}_{i}$

#### 5.2.3. Comparison of ${h}_{m}$ and ${h}_{\tau}$

#### 5.2.4. Empirical Constants ${D}_{b}$, ${E}_{b}$, ${D}_{i}$ and ${E}_{i}$

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Schematic of the experimental flume, the dark blue denoting the flow in the flume and (

**b**) images of the experimental site.

**Figure 4.**Comparison of the measured and analytical velocities. Black squares denote the measured data and red lines denote the analytical ones.

**Figure 5.**Comparison of the measured and analytical Reynolds stresses. Black squares denote the measured data and red lines denote the analytical ones.

**Figure 6.**Comparison of the measured and analytical turbulence intensity. Black squares denote the measured data and red lines denote the analytical ones.

**Table 1.**Basic details of the characteristic parameters. $B$ is the width of the flume, $H$ is the water depth, $S$ is the bed slope and $Re$ is the Reynolds number.

Cases | Cover Condition | $\mathit{B}$ (m) | $\mathit{H}$ (m) | $\mathit{S}$ | $\mathit{R}\mathit{e}$ |
---|---|---|---|---|---|

1 | Full ice cover | 1 | 0.15 | 0.001 | 25590 |

2 | Full ice cover | 1 | 0.185 | 0.001 | 33725 |

3 | Symmetrical shore cover | 1 | 0.16 | 0.001 | 32480 |

4 | Symmetrical shore cover | 1 | 0.20 | 0.001 | 49160 |

5 | Asymmetrical shore cover | 1 | 0.16 | 0.001 | 31888 |

**Table 2.**Error statistics for the longitudinal velocity, Reynolds stress and turbulence intensity. $\overline{{\Delta}_{a}}$ is the average absolute error and $\overline{{\Delta}_{r}}$ is the average relative error.

Cases | Velocity | Reynolds Stress | Turbulence Intensity | |||
---|---|---|---|---|---|---|

$\overline{{\Delta}_{\mathit{a}}}(\mathbf{m}/\mathbf{s})$ | $\overline{{\Delta}_{\mathit{r}}}(\%)$ | $\overline{{\Delta}_{\mathit{a}}}(\mathbf{m}/\mathbf{s})$ | $\overline{{\Delta}_{\mathit{r}}}(\%)$ | $\overline{{\Delta}_{\mathit{a}}}(\mathbf{m}/\mathbf{s})$ | $\overline{{\Delta}_{\mathit{r}}}(\%)$ | |

1 | 0.0065 | 3.99 | 0.036 | 7.01 | 0.0013 | 10.52 |

2 | 0.0057 | 3.18 | 0.038 | 10.00 | 0.0011 | 8.09 |

3 | 0.0059 | 2.86 | 0.050 | 9.11 | 0.0012 | 8.05 |

4 | 0.011 | 5.25 | 0.082 | 8.42 | 0.0012 | 6.99 |

5 | 0.019 | 10.97 | 0.093 | 13.63 | 0.0019 | 12.54 |

Cases | ${\mathit{n}}_{\mathit{b}}$ | ${\mathit{n}}_{\mathit{i}}$ | ${\mathit{m}}_{\mathit{b}}$ | ${\mathit{m}}_{\mathit{i}}$ | ${\mathit{h}}_{\mathit{m}}/\mathit{H}$ | ${\mathit{h}}_{\mathit{\tau}}/\mathit{H}$ | ||
---|---|---|---|---|---|---|---|---|

Calculated | Measured | Calculated | Measured | |||||

1 | 0.013 | 0.018 | 6.35 | 4.84 | 0.43 | 0.56 | 0.37 | 0.35 |

2 | 0.012 | 0.017 | 7.13 | 5.31 | 0.43 | 0.58 | 0.43 | 0.44 |

3 | 0.015 | 0.017 | 5.41 | 4.96 | 0.48 | 0.42 | 0.43 | 0.43 |

4 | 0.014 | 0.02 | 6.57 | 4.30 | 0.40 | 0.44 | 0.39 | 0.44 |

5 | 0.015 | 0.019 | 5.63 | 4.59 | 0.45 | 0.54 | 0.39 | 0.39 |

Cases | ${\mathit{D}}_{\mathit{b}}$ | ${\mathit{E}}_{\mathit{b}}$ | ${\mathit{D}}_{\mathit{i}}$ | ${\mathit{E}}_{\mathit{i}}$ |

1 | 2.22 | 1.44 | 2.06 | 1.82 |

2 | 2.21 | 1.46 | 2.13 | 1.87 |

3 | 2.22 | 1.73 | 2.24 | 1.50 |

4 | 2.15 | 1.00 | 2.21 | 1.83 |

5 | 2.31 | 1.81 | 2.16 | 1.51 |

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**MDPI and ACS Style**

Zhang, J.; Wang, W.; Li, Z.; Li, Q.; Zhong, Y.; Xia, Z.; Qiu, H.
Analytical Models of Velocity, Reynolds Stress and Turbulence Intensity in Ice-Covered Channels. *Water* **2021**, *13*, 1107.
https://doi.org/10.3390/w13081107

**AMA Style**

Zhang J, Wang W, Li Z, Li Q, Zhong Y, Xia Z, Qiu H.
Analytical Models of Velocity, Reynolds Stress and Turbulence Intensity in Ice-Covered Channels. *Water*. 2021; 13(8):1107.
https://doi.org/10.3390/w13081107

**Chicago/Turabian Style**

Zhang, Jiao, Wen Wang, Zhanbin Li, Qian Li, Ya Zhong, Zhaohui Xia, and Hunan Qiu.
2021. "Analytical Models of Velocity, Reynolds Stress and Turbulence Intensity in Ice-Covered Channels" *Water* 13, no. 8: 1107.
https://doi.org/10.3390/w13081107